I know that all Banach spaces are (Hausdorff)locally convex spaces. I would like to verify that the converse is not true by giving an example of a space which is locally convex but not Banach. I am looking at the linear space $C([0,1],\left\|\cdot\right\|_1)$ of complex-valued functions defined on the compact interval $[0,1]$, where the norm is given by $$\left\|f\right\|_1=\int_{0}^{1}|f(t)|dt $$ for any $f\in C[0,1]$. The normed space $C([0,1],\left\|\cdot\right\|_1)$ is known to be not Banach. I don't have any idea if the said space is a locally convex space. Any tips? Thanks in advance...

  • 3
    $\begingroup$ Every normed space is Hausdorff and locally convex. A Banach space is a complete normed space. If you are looking at normed spaces, to be "non-Banach" is the same as to be incomplete. $\endgroup$ – Willie Wong Feb 1 '13 at 14:01
  • $\begingroup$ Maybe you can add the definition of local convexity you are trying to verify and add some thoughts (or a proof) why Willie's first sentence is correct. $\endgroup$ – Martin Feb 1 '13 at 14:18

This is sasisfied by the Schawrtz space, which is locally convex but isn't Banach. There is a class of spaces called Frechet spaces that are locally convex and not necessarily Banach. Wikipedia has a few examples.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.